<div class="profile" id="userprofile">
<div class="miniBioSegment">
<img class="miniProfileImage" id="userimage" src="http://dummyimage.com/100x100/000088/ffffff.png&text=Profile+image">
</div>
<div id="miniBio" class="miniBioSegment">
This is basic information about this person that you clicked.
</div>
<div id="miniQuote" class="miniBioSegment">
This is a tag line from the person that you clicked.
</div>
</div>

You should use inline-block instead of inline for more control. I used a width of 33%-2px because the browser rounds the div's size up therefore leading to overflowing. Your 5px margins weren't helping with the sum either.

<div class="profile" id="userprofile">
<div class="miniBioSegment">
<img class="miniProfileImage" id="userimage" src="http://dummyimage.com/100x100/000088/ffffff.png&text=Profile+image">
</div>
<div id="miniBio" class="miniBioSegment">
This is basic information about this person that you clicked.
</div>
<div id="miniQuote" class="miniBioSegment">
This is a tag line from the person that you clicked.
</div>
</div>
<button onclick="showProfile()">View Profile</button>